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Internal Diffusion Limited Aggregation with Critical Branching Random Walks

Amine Asselah, Vittoria Silvestri, Lorenzo Taggi

TL;DR

This work studies BIDLA, a variant of Internal Diffusion Limited Aggregation driven by critical Branching Random Walks on $\mathbb{Z}^d$. It establishes a dimension-dependent shape result: a deterministic spherical limit shape exists for $d\ge 3$, while no such spherical shape theorem holds for $d\le 2$, with polynomial-type inner and outer deviation bounds in higher dimensions. The authors develop a novel Random Barrier Growth technique to obtain outer bounds, and combine BRW second-moment analyses with Green’s-function estimates to control growth inside balls and on boundaries, yielding a full shape theorem in $d\ge 3$ and a rigorous no-go result in low dimensions. The results illuminate a phase transition in growth dynamics under branching diffusion and introduce tools applicable to related growth models such as ARW and DLA, with implications for understanding fluctuation scales and universality in stochastic growth processes.

Abstract

Internal Diffusion Limited Aggregation is an interacting particle system that describes the growth of a random cluster governed by the boundary harmonic measure seen from an internal point. Our paper studies IDLA in $\mathbb{Z}^d$ driven by critical branching random walks. We prove that, unlike classical IDLA, this process exhibits a phase transition in the dimension. More precisely, we establish the existence of a spherical shape theorem in dimension $d\geq 3$ and the absence of a spherical shape theorem for $d \leq 2$. Our bounds on the inner and outer worst deviations are of polynomial nature, which we expect to be a feature of this model.

Internal Diffusion Limited Aggregation with Critical Branching Random Walks

TL;DR

This work studies BIDLA, a variant of Internal Diffusion Limited Aggregation driven by critical Branching Random Walks on . It establishes a dimension-dependent shape result: a deterministic spherical limit shape exists for , while no such spherical shape theorem holds for , with polynomial-type inner and outer deviation bounds in higher dimensions. The authors develop a novel Random Barrier Growth technique to obtain outer bounds, and combine BRW second-moment analyses with Green’s-function estimates to control growth inside balls and on boundaries, yielding a full shape theorem in and a rigorous no-go result in low dimensions. The results illuminate a phase transition in growth dynamics under branching diffusion and introduce tools applicable to related growth models such as ARW and DLA, with implications for understanding fluctuation scales and universality in stochastic growth processes.

Abstract

Internal Diffusion Limited Aggregation is an interacting particle system that describes the growth of a random cluster governed by the boundary harmonic measure seen from an internal point. Our paper studies IDLA in driven by critical branching random walks. We prove that, unlike classical IDLA, this process exhibits a phase transition in the dimension. More precisely, we establish the existence of a spherical shape theorem in dimension and the absence of a spherical shape theorem for . Our bounds on the inner and outer worst deviations are of polynomial nature, which we expect to be a feature of this model.
Paper Structure (31 sections, 31 theorems, 172 equations, 3 figures)

This paper contains 31 sections, 31 theorems, 172 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Statement of results
  3. Sketch of the proof
  4. No shape theorem in low dimension
  5. Shape theorem in high dimension
  6. Related models and open questions
  7. Open questions
  8. Definition of BIDLA
  9. Branching Random Walks
  10. BRW restricted to a domain
  11. Branching IDLA
  12. Stabilization in a finite domain
  13. Preliminaries
  14. On Branching Random Walks
  15. Green's function estimates
  16. ...and 16 more sections

Key Result

Theorem 1.1

Let $(A(t))_{t\in \mathbb{N} }$ denote a BIDLA process on $\mathbb Z^d$ satisfying Assumption assumption.

Figures (3)

  • Figure 1: Four simulations of BIDLA clusters on $\mathbb{Z}^2$ at time $t=20000$, plotted together with a disc of radius $\sqrt{t/\pi}$. Here the different colours represent the arrival time of particles (where time is indexed by particle releases).
  • Figure 2: A depiction of the construction used in the proof.
  • Figure 3: A depiction of the events $E_1 , E_2 , E_3 , E_4$ defined above.

Theorems & Definitions (59)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 1.3
  • Remark 2.1
  • Lemma 2.2
  • Proposition 3.1: asselah2022time, Lemma 3.8
  • Proposition 3.2: asselah2022time, Theorem 1.3
  • Corollary 3.3
  • proof
  • Lemma 3.4: asselah2013logarithmic, Lemma 5.1
  • ...and 49 more